Instead, our approach is the following:
we design a candidate total order \(<\)  on \(T\) , computable in polynomial time.
If the bi-relation \((T,<)\)  has small twin-width, we are done, as we have found an order witnessing this fact.
However, if \((T,<)\)  has large twin-width, then its adjacency matrix w.r.t. \(<\) 
must contain a large high-rank grid by [1]}.
We then extract a subtournament \(T^{\prime } \subset T\)  in which the adjacency matrix w.r.t. \(<\) 
still has a substantial (but logarithmically smaller) high-rank grid,
and on which \(<\)  is roughly described by a FO transduction.
This is enough to witness that \(T\)  has large twin-width.
With some Ramsey arguments, we are also able to extract from \(T^{\prime }\) 
some obstruction from one of the classes \(\mathcal {F}_=,\mathcal {F}_\leqslant ,\mathcal {F}_\geqslant \) .
